cgeneric_pc_prec_correl: Build an 'inla.cgeneric' to implement the PC-prior of a...

In graphpcor: Models for Correlation Matrices Based on Graphs

Build an inla.cgeneric to implement the PC-prior of a precision matrix as inverse of a correlation matrix.

Description

Build an inla.cgeneric to implement the PC-prior of a precision matrix as inverse of a correlation matrix.

Usage

cgeneric_pc_prec_correl(
  n,
  lambda,
  theta.base,
  debug = FALSE,
  useINLAprecomp = TRUE,
  libpath = NULL
)

Arguments

n	integer to define the size of the matrix
lambda	numeric (positive), the penalization rate parameter
theta.base	numeric vector with the model parameters at the base model
debug	integer, default is zero, indicating the verbose level. Will be used as logical by INLA.
useINLAprecomp	logical, default is TRUE, indicating if it is to be used the shared object pre-compiled by INLA. This is not considered if 'libpath' is provided.
libpath	string, default is NULL, with the path to the shared object.

Details

The precision matrix parametrization step 1:

Q0 = \left[ \begin{array}{ccccc} 1 & & & & \\ \theta_1 & 1 & & & \\ \theta_2 & \theta_n & & & \\ \vdots & & \ldots & \ddots & \\ \theta_{n-1} & \theta_{2n-3} \ldots & \theta_m & 1 \end{array} \right]

step 2: V = Q0^{-1}

step 3: S = diag(V)^{1/2}

step 4: C = SVS

step 5: Q = C^{-1}

p(Q|\lambda) = p(\theta[1:m] | lambda) =

p_C(C(Q)) | Jacobian C(Q) |

where p_C is the PC-prior for correlation, see section 6.2 of Simpson et. al. (2017), which is based on the hypersphere decomposition.

The hypershere decomposition, as proposed in Rapisarda, Brigo and Mercurio (2007) consider \theta[k] \in [0, \infty], k=1,...,m=n(n-1)/2 compute x[k] = pi/(1+exp(-theta[k])) organize it as a lower triangle of a n \times n matrix

B[i,j] = \left\{\begin{array}{cc} cos(x[i,j]) & j=1 \\ cos(x[i,j])prod_{k=1}^{j-1}sin(x[i,k]) & 2 <= j <= i-1 \\ prod_{k=1}^{j-1}sin(x[i,k]) & j=i \\ 0 & j+1 <= j <= n \end{array}\right.

Result

\gamma[i,j] = -log(sin(x[i,j]))

KLD(R) = \sqrt(2\sum_{i=2}^n\sum_{j=1}^{i-1} \gamma[i,j]

Value

a inla.cgeneric, cgeneric() object.

References

Daniel Simpson, H\aa vard Rue, Andrea Riebler, Thiago G. Martins and Sigrunn H. S\o rbye (2017). Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors Statistical Science 2017, Vol. 32, No. 1, 1–28. <doi 10.1214/16-STS576>

Rapisarda, Brigo and Mercurio (2007). Parameterizing correlations: a geometric interpretation. IMA Journal of Management Mathematics (2007) 18, 55-73. <doi 10.1093/imaman/dpl010>

graphpcor documentation built on June 8, 2025, 10:37 a.m.